Understanding fractions - GMAT Quantitative Reasoning

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Question

If $\dpi{100} \small x=\frac{1}{2}$ and $\dpi{100} \small y=\frac{1}{3}$ , which of the following is the smallest?

Answer

It can be solved by calculating all five answers:

$\dpi{100} \small x+y = \frac{5}{6}$

$\dpi{100} \small x-y=\frac{1}{6}$

$\dpi{100} \small (xy)^{2}=\frac{1}{36}$

$\dpi{100} \small x^{2}+y^{2}=\frac{13}{36}$

$\dpi{100} \small (x+y)^{2}=\frac{25}{36}$

The smallest is $\dpi{100} \small \frac{1}{36}$ .

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Question

What is $\dpi{100} \small 66\frac{2}{3}$ % of 18?

Answer

We need to convert this percentage into a fraction. This is one of the conversions you should remember.

$\dpi{100} \small 66\frac{2}{3}$ % = 0.666666 = $\dpi{100} \small \frac{2}{3}$

$18 \ast \frac{2}{3} = 6 \ast 2 = 12$

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Question

Which of the following is less than $\small \frac{3}{8}$ ?

Answer

It's easiest to convert the fractions into decimals.

$\small \frac{3}{8}\ =\ 0.375$

$\small \frac{2}{4}\ =\ 0.50$

$\small \frac{2}{5}\ =\ 0.40$

Therefore, the correct answer is 0.25.

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Question

Given that $\small 4 \leq A \leq 5$ and $\small \small 2 \leq B \leq 3$ , what is the range of possible values for ?

Answer

To get the smallest possible , subtract the greatest possible from the smallest possible ; this is .

To get the greatest possible , subtract the smallest possible from the greatest possible ; this is .

$\small 1 \leq A -B \leq 3$

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Question

If $\small x + y = 10$ and $\small xy = 20$ , then evaluate $\small x-y$ .

Answer

$\small x + y = 10$

$\small (x + y )^{2} = 10^{2}$

$\small x^{2} + 2xy + y^{2} = 100$

$\small \small x^{2} + 2xy + y^{2} - 4xy= 100 - 4xy$

$\small \small x^{2} - 2xy + y^{2} = 100 -4xy$

$\small (x - y)^{2} = 100 - 4 (20)$

$\small \small (x - y)^{2} = 20$

Either $\small \small \small x - y = \sqrt{20} = 2 \sqrt{5}$ or $\small \small x - y = - \sqrt{20} = - 2 \sqrt{5}$

But without further information, it is impossible to tell which is true. Therefore, the correct choice is that it cannot be determined from the information given.

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Question

Galactic Bounty Hunters, Inc has two departments: Trainees and Veterans. If on an average week, the each member of the Trainee department arrests $\frac{3}{5}$ as many criminals as each member of the Veteran department, but the Veteran department has $\frac{1}{3}$ as many members as the Trainee department, what fraction of the arrests were made by the members of the Veteran department?

Answer

What we want to do is pick numbers for the number of arrests made during the week and for the number of members in each department.

Let's take the number of arrests first. Let's say each Trainee arrests 3 criminals. Then, since the Trainees make 3/5 the number of arrests the Veterans make, we have:

$\frac{3}{5}v_a=3$

$v_a=3\bigg(\frac{5}{3}\bigg)=5$

So, each Veteran would arrest 5 criminals.

Next, we know that there are 1/3 as many Veterans as Trainees. There if we have 3 Trainees, then we have 1 Veteran.

Using this information we can create the following equation for total arrests made by each department:

$\frac{t_a(t)}{v_a(v)}$

Where is the number of Trainee arrests times the number of Trainees and is the number of Veteran arrests times the number of Veterans

$\frac{t_a(t)}{v_a(v)}=\frac{3(3)}{5(1)}=\frac{9}{5}$

We're almost done. Since we have 3 Trainees, and each arrests 3 criminals, the total number of Trainee arrests is 9.

Since we have 1 Veteran, and each arrests 5 criminals, the total number of Veteran arrests is 5.

The total number of arrests is

The fraction of the arrests made by the Veterans is:

$\frac{5}{14}$

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Question

Find the result and simplify the following expression: $\frac{1}{1-\frac{2}{5}}+ \frac{1}{1+\frac{1}{5}}$

Answer

We start by simplifying the denominators:

$\frac{1}{1-\frac{2}{5}}=\frac{1}{\frac{5}{5}-\frac{2}{5}} = \frac{1}{\frac{3}{5}}$ and $\frac{1}{1+\frac{1}{5}} = \frac{1}{\frac{5}{5}+\frac{1}{5}}=\frac{1}{\frac{6}{5}}$

We know that: $\frac{1}{\frac{3}{5}}+\frac{1}{\frac{6}{5}} = \frac{5}{3}+\frac{5}{6}$

Then we put both fractions to the same denominator, and don't forget to simplify the fraction:

$\frac{5}{3}+\frac{5}{6}=\frac{10}{6}+\frac{5}{6}=\frac{15}{6}=\frac{5}{2}$

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Question

Simplify the following into a single fraction.

$\frac{a}{b} + \frac{6a}{7c}$

Answer

To simply, we must first find the common denominator of the two fractions. That would be $b \times 7c$ or

Hence we multiply the first fraction by $\frac{7c}{7c}$ and the second fraction by $\frac{b}{b}$ , and we will have.

$\frac{7ac}{7bc} + \frac{6ab}{7bc}$ .

Now that the denominators match, we can add the fractions. The denominator stays the same after this, only the numerators add together.

$\frac{7ac+6ab}{7bc}$

Then factor out an from the numerator to get the final answer.

$\frac{a(7c+6b)}{7bc}$

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Question

Which of the following is false?

Answer

$\frac{a}{b} +\frac{c}{d} = \frac{a+c}{bd}$ is false because is not displaying the correct way to add two fractions together. When adding fractions we must find a common denominator before adding the numerators.

For example, if , then the above expression would read

$\frac{1}{2} +\frac{1}{2} = \frac{1}{2}$ This we know is absurd!

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Question

What is the least common denominator of the following fractions?

$\frac{7}{5}, \frac{8}{15},\frac{2}{3}$

Answer

The least common denominator (LCD) is the lowest common multiple of the denominators.

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

Multiples of 15: 15, 30, 45, 60, 75, 90,105,120, 135, 150

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

The least common denominator is 15.

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Question

Simplify the following expression: $\frac{5}{6}+\frac{7}{11}+\frac{4}{3}$

Answer

In order to simplify the expression, we start by finding the common denominator of each term in $\frac{5}{6}+\frac{7}{11}+\frac{4}{3}$ :

In this case the common denominator is 66.

$\frac{5}{6}\left(\frac{11}{11} \right )+\frac{7}{11}\left(\frac{6}{6} \right )+\frac{4}{3}\left(\frac{22}{22} \right )$

$=\frac{55}{66}+\frac{42}{66}+\frac{88}{66}$

$=\frac{55+42+88}{66}$

$=\frac{185}{66}$

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Question

What value must take in order for the following expression to be greater than zero?

$\frac{3}{k}-\frac{5}{7}$

Answer

is such that:

$\frac{3}{k}-\frac{5}{7}>0$

Add $\frac{5}{7}$ to each side of the inequality:

$\frac{3}{k}>\frac{5}{7}$

Multiply each side of the inequality by :

$\frac{21}{k}>5$

Multiply each side of the inequality by :

Divide each side of the inequality by :

$k< \frac{21}{5}$

You can now change the fraction on the right side of the inequality to decimal form.

$k<\frac{21}{5}=\frac{20}{5}+\frac{1}{5}= 4.2$

The correct answer is , since k has to be less than for the expression to be greater than zero.

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Question

$\frac{2}{3}+\frac{3}{4}+\frac{5}{6}=$

Answer

In order to add these fractions, we must first find a common denominator for all three. Looking at our three different denominators, we can see that their lowest common denominator is 12. Therefore, we need to rewrite each fraction with a denominator of 12:

$\frac{2}{3}+\frac{3}{4}+\frac{5}{6}=\frac{2(4)}{3(4)}+\frac{3(3)}{4(3)}+\frac{5(2)}{6(2)}=\frac{8}{12}+\frac{9}{12}+\frac{10}{12}$

Now that the fractions all have a common denominator, we can simply add them together and then simplify the result, recognizing that 27 and 12 have a common factor of 3:

$\frac{8}{12}+\frac{9}{12}+\frac{10}{12}=\frac{27}{12}=\frac{9}{4}$

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Question

Which of the following is less than $\frac{5}{8}$ ?

Answer

In order to determine which fractions are less than $\frac{5}{8}$ , it's best to convert them to decimals and compare:

$\frac{5}{8}=0.625$

$\frac{1}{2}=0.5$

$\frac{3}{4}=0.75$

$\frac{2}{3}=0.67$

$\frac{5}{6}=0.83$

$\frac{7}{6}=1.167$

Therefore, $\frac{1}2{}$ is the only choice that is less than $\frac{5}{8}$ .

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Question

Simplify the following into a single fraction: $\frac{x}{y}+\frac{ab}{cd}$

Answer

In order to simplify $\frac{x}{y}+\frac{ab}{cd}$ , combine the terms by finding the common denominator:

$\frac{x}{y}+\frac{ab}{cd}$

$=\frac{x(cd)}{y(cd)}+\frac{(y)ab}{(y)cd}$

$=\frac{xcd+yab}{ycd}$

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Question

Define an operation $\ast$ as follows:

For all real numbers ,

$a \ast b = \frac{1}{4}a + b^{2}$

Evaluate $\frac{2}{3} \ast \frac{1}{3}$ .

Answer

$a \ast b = \frac{1}{4}a + b^{2}$

$\frac{2}{3} \ast \frac{1}{3}= \frac{1}{4} \cdot \frac{2}{3} + \left (\frac{1}{3} \right ) ^{2}$

$= \frac{2}{12} + \frac{1}{9}$

$= \frac{1}{6} + \frac{1}{9}$

$= \frac{3}{18} + \frac{2}{18}$

$= \frac{5}{18}$

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Question

Casius has $\tiny \frac{1}{6}$ of a liter of water. Olivar has $\tiny \frac{4}{7}$ of the amount Casius has. How many liters of water does Olivar have?

Answer

This question tests your ability to read word problems and multiply fractions. Because we know that Olivar has $\frac{4}{7}$ of what Casius has, we can see that we need to multiply. Perform the following operation to find the answer.

$\frac{1}{6}*\frac{4}{7}=\frac{4}{42}=\frac{2}{21}$

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Question

Define an operation $\ast$ as follows:

For all real numbers ,

$a \ast b = 2a \div b$

Evaluate $5\frac{1}{6} \ast 3\frac{1}{3}$ , and round to the nearest whole number.

Answer

$a \ast b = 2a \div b$

$5\frac{1}{6} \ast 3\frac{1}{3} = 2 \left ( 5\frac{1}{6} \right ) \div 3\frac{1}{3}$

$= 2 \left ( \frac{31}{6} \right ) \div \frac{10}{3}$

$= \frac{31}{3} \div \frac{10}{3}$

$= \frac{31}{3} \cdot \frac{3}{10}$

$= \frac{31} {10}$

$=3 \frac{1} {10}$

This rounds to 3.

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Question

Seven thirds of eighteen seventieths is what?

Answer

Seven thirds of eighteen seventieths is

$\frac{7}{3} \times \frac{18 }{70} = \frac{7}{3} \times \frac{9}{35} = \frac{1}{1} \times \frac{3}{5}= \frac{3}{5}$

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Question

One orange yields $\frac{1}{3}$ cup of juice; one gallon is equal to 16 cups. How many oranges are needed to yield one half gallon of orange juice?

Answer

One half gallon comprises 8 cups. To find the number of oranges needed, divide:

$8 \div \frac{1}{3} = 24$

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